Probability density,diagrammatic technique,and epsilon expansion in the theory of wave turbulence

We apply the methods of field theory to study the turbulent regimes of statistical systems. First we show how one can find their probability densities. For the case of the theory of wave turbulence with four-wave interaction we calculate them explicitly and study their properties. Using those densities we show how one can in principle calculate any correlation function in this theory by means of direct perturbative expansion in powers of the interaction. Then we give the general form of the corrections to the kinetic equation and develop an appropriate diagrammatic technique. This technique, while resembling that of ϕ⁴ theory, has many new distinctive features. The role of the ϵ = d − 4 parameter of ϕ⁴ theory is played here by the parameter κ = β + d − α − γ where β is the dimension of the interaction, d is the space dimension, α is the dimension of the energy spectrum and γ is the “classical” wave density dimension. If κ > 0 then the Kolmogorov index is exact, and if κ < 0 then we expect it to be modified by the interaction. For κ a small negative number, α < 1 and a special form of the interaction we compute this modification explicitly. We neglect the encountered IR divergencies with the intend to study them in a later publication.